25. (a) Find all 2 ×x 2 matrices whose null space is the line3x – 5y = 0.(b) Describe the null spaces of the following matrices:6 2C =3AB =D =516. (Calculus required) Show that the following sets of functionsare subspaces of F(-∞, ∞).(a) All continuous functions on (–0∞, 00).(b) All differentiable functions on (-∞, ).(c) All differentiable functions on (–o, 0) that satisfyf'+ 2f = 0.9. Find the dimension of each of the following vector spaces.(a) The vector space of all diagonal n x n matrices.(b) The vector space of all symmetric n x n matrices.(c) The vector space of all upper triangular n x n matrices.

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25. (a) Find all 2 × 2 matrices whose null space is the line 3x – 5y = 0. (b) Describe the null spaces of the following matrices: |6 2 C = D = A = B = 5

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 26EQ

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Question

25. (a) Find all 2 ×x 2 matrices whose null space is the line
3x – 5y = 0.
(b) Describe the null spaces of the following matrices:
6 2
C =
3
A
B =
D =
5
16. (Calculus required) Show that the following sets of functions
are subspaces of F(-∞, ∞).
(a) All continuous functions on (–0∞, 00).
(b) All differentiable functions on (-∞, ).
(c) All differentiable functions on (–o, 0) that satisfy
f'+ 2f = 0.
9. Find the dimension of each of the following vector spaces.
(a) The vector space of all diagonal n x n matrices.
(b) The vector space of all symmetric n x n matrices.
(c) The vector space of all upper triangular n x n matrices.

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